\chapter{Equations}

The ideal gas law is given as

\begin{equation}
p=\rho \mathcal{R} T/M.
\label{eq:Ideal gas}
\end{equation}

Where M is the molecular mass of the mixture.

The 1st law of thermodynamics for an open system when kinetic and potential energy is ignored
%First Law of Thermodynamics
\begin{equation}
\frac{dE_{cv}}{dt}=\dot{Q}-\dot{W}+\sum_i{\dot{m}_i h_i}-\sum_e{\dot{m}_e h_e}.
\label{eq:1st law}
\end{equation}

where $i$ is the flow entering the system and $e$ is the flow exiting the system

The second law of thermodynamics is given as
%Second Law of Thermodynamics
\begin{equation}
\frac{dS_{cv}}{dt}=\sum_j{\frac{\dot{Q}_j}{T_j}}+\sum_i{\dot{m}_i s_i}-\sum_e{\dot{m}_e s_e}+{\dot{\sigma}_{cv}}.
\label{eq:2nd law}
\end{equation} 


%heat capacity
The specific heat capacity is defined as
\begin{equation}
c_p=\left(\frac{\partial h}{\partial T} \right)_p
\label{eq:Cp}
\end{equation}

The specific heat capacity of the mixture,$c_p$ is given by
\begin{equation}
c_p=\sum_k{Y_k c_{p,k}}.
\label{eq:sum_cp}
\end{equation}

For an ideal mixture of ideal gases
%Total enthalpy
\begin{equation}
h=\sum_k{Y_k h_k (T)}
\label{eq:totalh}
\end{equation}

\begin{equation}
s=\sum_k{Y_k s_k(T,p_k)}.
\label{eq:ideal mixture entropy}
\end{equation}


The chemical potential for an ideal gas is given as

%Chemical potential for ideal gas
\begin{equation}
\mu_k=g_k=h_k-T s_k=h_k(T)-Ts^o_k(T)+\frac{\mathcal{R}}{M_k} ln \frac{p_k}{p_{ref}}.
\label{eq:chempot}
\end{equation}

The absolute entropy of an ideal gas is given as
%Absolute entropy of an ideal gas
\begin{multline}
s_k = s_k(T,p_k) =s^o_k - \int_{p_{ref}}^{p_k} {\mathcal{R}_k \frac{dp}{p}}=\\s_k(T_{ref},p_{ref})+\int_{T_{ref}}^T {\left(\frac{c_{p,k}(T)}{T} \right)_{p_{ref}} dT}-\mathcal{R}_k ln \frac{p}{p{ref}}.
\label{eq:absentropy}
\end{multline}

From \cite[p.540]{MS} 

\begin{equation}
T dS=dE +p dV - \sum_k{\overline{\mu}_k dn_k}.
\label{eq:Tds }
\end{equation}

The thermal conductivity of a mixture of gases can be approximated by the following relation, within an error of $\pm 10-20\%$ as found in \cite{warnatz2006combustion}:
\begin{equation}
\lambda=\frac{1}{2} \left[ \sum_{i}{X_i \lambda_i}+\left( \sum_{i}{\frac{Y_i}{\lambda_i}} \right)^{-1} \right].
\label{eq:conductivity of mixture}
\end{equation}

For the simulations the following curvefits where made with the method described in \ref{sec:curve fitting} from Table A.4 in Incropera and Dewitt's \textit{Fundamentals of Heat and Mass Transfer} \cite{incropera2007fundamentals}:
\begin{eqnarray*}
\lambda_{H_2}(T)= 3.2778e-008T^2+0.00032748T+0.090588 &\text{ for } 300<T<2000 K\\
\lambda_{N_2}(T)= 5.3835e-005T+0.011313 &\text{ for } 300<T<1300 K\\
\lambda_{O_2}(T)=6.0545e-005T+0.0097264 &\text{ for } 300<T<1300 K\\
\lambda_{H_2O(g)}(T)=1.3815e-008T^2+6.635e-005T-0.0026736 &\text{ for } 380<T<850\\
\label{eq:condcurvefits}
\end{eqnarray*}
